The t Distributions Sections 20.1, 20people.hsc.edu/faculty-staff/robbk/math121/lectures... · Robb...
Transcript of The t Distributions Sections 20.1, 20people.hsc.edu/faculty-staff/robbk/math121/lectures... · Robb...
The t DistributionsSections 20.1, 20.2
Lecture 35
Robb T. Koether
Hampden-Sydney College
Thu, Mar 24, 2016
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 1 / 18
Outline
1 The Central Limit Theorem
2 Substituting s for σ
3 The t Distributions
4 Comparison of t to z
5 Assignment
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 2 / 18
Outline
1 The Central Limit Theorem
2 Substituting s for σ
3 The t Distributions
4 Comparison of t to z
5 Assignment
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 3 / 18
The Central Limit Theorem
Theorem (The Central Limit Theorem)For any population, let its mean and standard deviation be µ and σ,respectively. let x be the sample mean of samples of size n. Then xhas an approximately normal distribution with mean µ and standarddeviation σ/
√n.
If the sample size is large enough (n ≥ 30), then theapproximation is good enough for applications.
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 4 / 18
The Central Limit Theorem
It follows from this theorem that the statistic z, defined as
z =x − µσ/√
n
has an approximately normal distribution.
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 5 / 18
Special Case
Theorem (Special Case of the CLT)If the population is normal, then for all sample sizes, no matter howsmall, x has an exactly normal distribution with mean µ and standarddeviation σ/
√n.
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 6 / 18
Special Case
It follows from this theorem that if the population is normal, thenthe statistic
z =x − µσ/√
n
has an exactly normal distribution for any sample size n.
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 7 / 18
Outline
1 The Central Limit Theorem
2 Substituting s for σ
3 The t Distributions
4 Comparison of t to z
5 Assignment
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 8 / 18
Substituting s for σ
In practice we rarely know the value of σ.However, we do know the value of s and s is an estimator of σ.What is the effect of replacing σ with s in the formula
x − µσ/√
n?
s is a variable and σ is a constant.The effect is to increase the variability, and therefore theuncertainty, of the value of the statistic.
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 9 / 18
Outline
1 The Central Limit Theorem
2 Substituting s for σ
3 The t Distributions
4 Comparison of t to z
5 Assignment
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 10 / 18
The t Statistic
It turns out that the statistic
x − µs/√
n
does not have a normal distribution, even when sampling from anormal population, unless n is fairly large (n > 100).So we name the statistic t instead of z:
t =x − µs/√
n.
Furthermore, the exactly distribution of t is different for differentsample sizes.(As n increases, t tends towards z.)
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 11 / 18
The t Distributions
Definition (The t Distributions)If we draw a simple random sample of size n from a normal populationwith mean µ and standard deviation σ, then the t statistic with n − 1degrees of freedom is
t =x − µs/√
n
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 12 / 18
Outline
1 The Central Limit Theorem
2 Substituting s for σ
3 The t Distributions
4 Comparison of t to z
5 Assignment
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 13 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
2 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
3 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
4 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
5 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
6 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
7 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
8 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
9 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
Comparison of t to z
-3 -2 -1 1 2 3
0.1
0.2
0.3
0.4
10 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 14 / 18
At n = 10, the difference between t and z does not appear to begreat.However, p-values usually involve the tails.How do the upper tails of t and z compare?
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 15 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
2 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
3 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
4 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
5 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
6 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
7 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
8 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
9 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
10 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
30 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
50 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Comparison of the Upper Tails
2.5 3.0 3.5 4.0 4.5
0.01
0.02
0.03
0.04
200 degrees of freedom
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 16 / 18
Outline
1 The Central Limit Theorem
2 Substituting s for σ
3 The t Distributions
4 Comparison of t to z
5 Assignment
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 17 / 18
Assignment
AssignmentRead Sections 20.1, 20.2.Apply Your Knowledge: 1, 3, 4.Check Your Skills: 17, 18, 19.
Robb T. Koether (Hampden-Sydney College) The t DistributionsSections 20.1, 20.2 Thu, Mar 24, 2016 18 / 18